ARITHMETIC CIRCUITS

Arithmetic Circuitscan be defined as the standard model used for computingpolynomials. Anarithmetic circuituses variables or numbers as inputs and can either add or multiply two expressions it had computed earlier.Arithmetic circuitsprovide a way to understand better the complexity in regard to computingpolynomials.

We can use an arithmetic model to illustrate this. Anarithmetic circuitΦ over the field F and the set of variables X (usually, X = {x1, . . . , xn}) is a directed acyclic graph as follows. The vertices of Φ are called gates. Every gate in Φ of in-degree 0 is labeled by either a variable from X or a field element from F. Every other gate in Φ is labeled by either × or + and has in-degree 2. Anarithmetic circuitis called a formula if it is a directed tree whose edges are directed from the leaves to the root.

Every in-degree 0 gate is called an input gate, even in those instances when the gate is labeled by a field element. Every gate of out-degree 0 is called an output gate. Every gate labeled by × is called a product gate and every gate labeled by + is called a sum gate. The size of Φ, denoted |Φ|, is the number of edges in Φ. The depth of a gate v in Φ denoted as depth (v), is the length of the longest directed path reaching v. The depth of Φ is the maximal depth of a gate in Φ. When speaking of bounded depth circuits, circuits whose depth is bounded by a constant independent of |X| — we do not have a restriction on the fan-in. For two gates u and v in Φ, if (u,v) is an edge in Φ, then u is called a child of v, and v is called a parent of u.

Anarithmetic circuitcomputes apolynomialin a natural way: An input gate labeled by α ∈ F ∪ X computes thepolynomialα. A product gate computes the product of thepolynomialscomputed by its children. A sum gate computes the sum of thepolynomialscomputed by its children.

For a gate v in Φ, define as Φv to be the sub-circuit of Φ rooted at v. Denote by Xv the set of variables that occur in the circuit Φv. We usually denote by fv thepolynomialin F[Xv] computed by the gate v in Φ. We sometimes abuse notation and denote by Φv thepolynomialcomputed by v as well. Define the degree of a gate v, denoted deg(v), to be the total degree of thepolynomialfv. The degree of Φ is the maximal degree of a gate in Φ.

Everypolynomialf ∈ F[X] can be computed by an arithmetic circuit and by an arithmetic formula. We only need to know how many gates are needed for the computation.

The above definition shows an existing difference betweenarithmetic circuitsand Boolean circuits. Arithmetic circuits cannot perform operations on the “bit representation” of the input field elements that are not arithmetic operations but a Boolean circuit can do the same.

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